Statistics Chapter 3: Central Tendency

Chapter 3: Central Tendency

• Central tendency measures the center of a distribution. Examples include mean, median, and mode.
• Scientists, therapists, and educators frequently need to understand the central tendency of data. For example, finding the average number of symptoms in a disorder, or the most frequent symptom.

Mean

• The mean is the sum of scores divided by the number of scores. This is also known as the average.
• Population mean (μ): μ = Σx/N where μ is the Greek letter mu, Σx is the sum of all scores, and N is the population size.
• Sample mean (M): M = Σx/N where N is the size of the sample.
• The mean may be misleading if there are extreme values (outliers) in the data. For example, a psychotherapy school might claim a mean hourly rate of $500, but if a few psychotherapists charge $2,100/hour and the others are in the $100 range, the average will be inflated. The median will be more accurate in cases like this.

Median

• The median is the middle number in an ordered set of numbers.
• For odd-sized sets, it is the middle value after ordering the data. For even-sized sets it is the average of the two middle values.
• Example: Find the median of the odd set 1, 0, 5, 4, 6; Order the numbers: 0, 1, 4, 5, 6; The middle number (3rd number) is 4.
• Example: Find the median of the even set 2, 8, 0, 6, 4, 5; Order the numbers: 0, 2, 4, 5, 6, 8; The middle two numbers are 4 and 5; The average of these two numbers is 4.5.
• The median is less sensitive to outliers than the mean.

Mode

• The mode is the most frequently occurring score or value.
• Example: Find the mode for the data 1, 2, 2, 2, 3, 4. The mode is 2.
• Example: Find the mode for the data 1, 2, 2, 3, 4, 4. The modes are 2 and 4.
• Example: Find the mode for the data blue, blue, pink, pink, gray, gray, gray. The mode is gray.
• The mode can only be determined from nominal scale data.

Describing Distributions

• Graphed distributions can vary in skew (symmetry) and kurtosis (pointedness).
• Positively skewed distribution: the tail of the distribution trails off to the right.
• Negatively skewed distribution: the tail of the distribution trails off to the left.
• Normal distribution: a symmetrical distribution where the mean, median, and mode are all located at the center.

The Empirical Rule

• For normally distributed data:
  • Approximately 68% of the data falls within 1 standard deviation of the mean.
  • Approximately 95% of the data falls within 2 standard deviations of the mean.
  • Approximately 99.7% of the data falls within 3 standard deviations of the mean.

Chapter 4: Variability

• Variability describes how dispersed or spread out data points are.
• Range: the difference between the highest and lowest score. The range is helpful if there are no major outliers.
• Variance: the average squared distance that scores deviate from their mean.
• Standard Deviation: the average distance that scores deviate from their mean. (The standard deviation is the square root of the variance).
• The Empirical Rule is used to understand the proportion of data points falling within specific ranges of the mean using the standard deviation.